Further reading
Fourier and heat conduction
Jean-Baptiste Joseph Fourier (1768–1830) was born in Auxerre, France, the son of a tailor. Orphaned before 10 years of age, he was educated at the city’s military school, run by Benedictine monks. As a boy he wrote sermons that priests from other cities used, and at the 13 age he discovered mathematics, in which he quickly excelled. He prepared to enter the Benedictine order, but the Revolution of 1789 changed his plans.
Fourier enthusiastically joined the revolutionary cause—and was nearly guillotined in the Terror of 1794. With the creation of the Escola Normal and the Escola Politécnica in Paris, he became a professor and began the work that would make him famous; from this time is his method for separating the real roots of algebraic equations, a problem that Newton had initiated.
In 1798 he accompanied Napoleon on the expedition to Egypt , where he was secretary of the Cairo Institute, directed engineering work and became an expert in Egyptology. Back in France, he was appointed prefect of the Isère department, in Grenoble, and there, between 1804 and 1807, he studied the heat propagation in a solid body. To solve the equations he found, he wrote temperature as an infinite sum of sines and cosines — a type of series development different from Taylor’s, because it uses periodic functions instead of powers.
The idea that any functions, even those with “corners”, could be written in this way seemed absurd to Lagrange and other members of the Academy, who received the 1807 memoir with reservations. Fourier insisted, and in 1822 published the Théorie analytique de la chaleur , a landmark in mathematical physics. To the Fourier series They ended up demanding a complete revision of the concepts of function, convergence and integral — work to which Dirichlet, Riemann and Cantor dedicated themselves — and today they are everywhere: in the compression of music and images, in the analysis of earthquakes and in magnetic resonance. Fourier died in Paris in 1830.
The waves in this chapter are precisely the Fourier pieces: add $\operatorname{sen} x$, $\frac{1}{3}\operatorname{sen} 3x$, $\frac{1}{5}\operatorname{sen} 5x$, … and the curve becomes more and more similar to a square wave.
Partial sums of a Fourier series: with 1, 3 and 10 terms, the sum of sines approximates a square wave.
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Sources: Fourier (MacTutor) · J. Fourier, Théorie analytique de la chaleur (1822).
